Advanced Methods of Structural Analysis

8.3 Comparison of the Force and Displacement Methods 293

4.The determinant of the matrix of coefficients of the canonical equations is strictly
positive. This condition describes an internal property of structures based on a
fundamental law of elastic systems: the potential energy of a structure subjected
to any load is positive. SinceD¤ 0 , the solution to the canonical equations of
any redundant structure, subjected to any load, change of temperature, or settle-
ments of supports, is unique.

It is time to ask a question: when is it more convenient to apply the force method

and when the displacement method? The general answer is the following: the more

rigid the system due to given constraints, then the more efficient the displace-

ment method will be. This can be illustrated by considering different structures.

Figure8.9a presents a frame with fixed supports. The frame, according to the force

method, has nine unknowns, while by the displacement method it has only one un-

known: the angle of rotation of the only rigid joint. Analysis of this structure by

the displacement method is a very simple problem. Should this structure be modi-

fied by adding more elements connected at the rigid joint, nevertheless, the number

of unknowns by the displacement method isstill the same, while the number of

unknowns by the force method is increasedwith the addition of each new element.

P EI=∞

ab c

e

f

d

Fig. 8.9 Different types of structures that can be analyzed by either the force method or the dis-

placement method

Another frame is shown in Fig.8.9b. The number of unknowns by the force

method is one, while by the displacement method it is six (four rigid joints and

two linear independent displacements). Also, it is important to note that this frame

contains inclined members, which lead to additional cumbersome computations of

the reactions in the introduced constraints. Therefore the force method is more

preferable.

A statically indeterminate arch with fixed supports can also be analyzed by

both the force and displacement methods. The number of unknowns by the force

method is three. In order to analyze this kind of arch by the displacement method,

its curvilinear axis must be replaced by a set of straight members (since the stan-

dard elements of the displacement methodare straight members). One version of

such segmentation of the arch is shown in Fig.8.9c by dashed lines; in this case,

the number of unknowns by the displacement method is six. Obviously, the force

method is more preferable here. If the arch is part of any complex structure, as